Manchester Geometry Seminar - Dmitry Roytenberg
| Dates: | 8 December 2025 |
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| Times: | 15:00 - 16:00 |
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| What is it: | Seminar |
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| Organiser: | Department of Mathematics |
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| Who is it for: | University staff, External researchers, Current University students |
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Speaker: Dmitry Roytenberg (Amsterdam)
Title: On dg-manifolds with vanishing Atiyah class
Abstract:
Differential graded (dg) manifolds are a generalization of manifolds comprising and combining (higher) Lie algebroids, on the one hand, and derived manifolds on the other. DG manifolds are objects of a homotopy theory, in the sense of there being a natural notion of equivalence between them that is weaker than that of an isomorphism (but reducing to isomorphism when the dg-structure is trivial). In this context it is natural to look for invariants of dg-manifolds under the weak equivalence. Among suitable candidates for such invariants are characteristic classes, of which one of the best known examples is the Atiyah class, which obstructs the existence of an affine connection compatible with the dg-structure. This class was first defined for general dg-manifolds by Shoikhet and has since then been studied by a number of authors, particularly in connection with deformation quantization and gauge theory. It turns out that the vanishing of the Atiyah class -- that is, the existence of a compatible connection -- imposes stringent regularity conditions on the dg-manifold in question. In particular, such a dg-manifold is necessarily, in a suitable sense, locally trivial, so that the global condition becomes purely topological (in case of foliations, one recovers Molino's results of the early 1970's). This talk is based on joint work in progress with Thanasis Chatzistavrakidis and Lara Jonke.
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Frank Adams Room 1
Alan Turing Building
Manchester
